Berry-Esséen bounds and almost sure CLT for the quadratic variation of a class of Gaussian process

We propose a condition which is valid for a class of continuous Gaussian processes that may fail to be self-similar or have stationary increments. Some examples include the sub-fractional Brownian motion and the bi-fractional Brownian motion and the sub-bifractional Brownian motion. Under this assumption, we show an upper bound for the difference between the inner product associated with a class of Gaussian process and that associated with the fractional Brownian motion. This inequality relates a class of Gaussian processes to the well studied fractional Brownian motion, which characterizes their relationship quantitatively. As an application, we obtain the optimal Berry-Esséen bounds for the quadratic variation when H∈(0,23] and the upper Berry-Esséen bounds when H∈(23,34]. As a by-product, we also show the almost sure central limit theorem (ASCLT) for the quadratic variation when H∈(0,34]. The results in the present paper extend and improve those in the literature.